Roots and Irrational Numbers Section 1 5 California

Roots and Irrational Numbers Section 1. 5

California Standards 2. 0 Students understand use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand use the rules of exponents.

Objectives: In this lesson you’ll: • Evaluate expressions containing roots. • Classify numbers within the real number system

Words to know… • Square root - a number which, when multiplied by itself, produces the given number. (Ex. 7² = 49, 7 is the square • root of 49) • Perfect square- any number that has an integer square root. (ex. 100 is a perfect square , • Cube root - a number that is raised to the third power to form a product is a cube root. (ex 23=8, =2)

Square Roots Squares 0² = 0 1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 Perfect Square Roots

Are squares and square roots inverses? A square root is the inverse operation of a square!

Do you know your perfect squares? 7 and -7 25 8 and -8 121 3 and -3 196

Square Roots Positive real numbers have two square roots. Find the square roots of 16. 4 4 = 42 = 16 (– 4) = (– 4)2 = 16 – =4 Positive square root of 16 = – 4 Negative square root of 16 The square roots of 16 are 4 and - 4.

You try Find each root. Think: What number squared equals 81? Think: What number squared equals 25? C. Think: What number cubed equals – 216? = – 6 (– 6)(– 6) = 36(– 6) = – 216

You try Finding Roots of Fractions. a. Think: What number squared equals b. Think: What number cubed equals

Words to know… • Natural numbers - The counting numbers. (example: 1, 2, 3…) • Whole numbers - The natural numbers and zero. (example: 0, 1, 2, 3…) • Integers -The whole numbers and their opposites. (ex: …-3, -2, 1, 0, 1, 2, 3…) • Rational numbers - Numbers that can be expressed as a fraction (a/b).

Words to know… • Terminating decimal -Rational numbers in decimal form that have finite (ends) number of digits. (ex 2/5= 0. 40 ) • Repeating decimal -rational numbers in decimal form that have a block for one or more digits that repeats continuously. (ex. 1. 3=1. 33333) • Irrational numbers - numbers that cannot be expressed as a fraction including square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat.

The real numbers are made up of all rational and irrational numbers. Reading Math Note the symbols for the sets of numbers. R: real numbers Q: rational numbers Z: integers W: whole numbers N: natural numbers

Classifying Real Numbers Write all classifications that apply to each real number. A. – 32 32 – 32 = – 1 – 32 = – 32. 0 – 32 can be written in the form . – 32 can be written as a terminating decimal. rational number, integer, terminating decimal B. irrational 14 is not a perfect square, so irrational. is

Check It Out! Write all classifications that apply to each real number. a. 7 7 4 can be written in the form 9 . can be written as a repeating decimal. 67 9 = 7. 444… = 7. 4 rational number, repeating decimal b. – 12 can be written in the form – 12 can be written as a terminating decimal. rational number, terminating decimal, integer .

Write all classifications that apply to each real number. irrational 10 is not a perfect square, so is irrational. 100 is a perfect square, so is rational. 10 can be written in the form and as a terminating decimal. natural, rational, terminating decimal, whole, integer

Lesson Quiz Find each square root. 1. 3 2. 3. 5 4. 1 5. The area of a square piece of cloth is 68 in 2. Estimate to the nearest tenth the side length of the cloth. 8. 2 in. Write all classifications that apply to each real number. 6. – 3. 89 rational, repeating decimal 7. irrational